96 An introduction to the physics of cosmology
We therefore expect that all scale-invariant models will have similar CMB
power spectra: a flat Sachs–Wolfe portion down toK1deg−^1 ,followedbya
bump where Doppler and adiabatic effects come in, which turns over on arcminute
scales through damping and smearing. This is illustrated well in figure 2.22,
which shows some detailed calculations of 2D power spectra, generated with
the CMBFAST package. From these plots, the key feature of the anisotropy
spectrum is clearly the peak at"∼100. This is often referred to as theDoppler
peak, but it is not so clear that this name is accurate. Our simplified analysis
suggests that Sachs–Wolfe anisotropy should dominate forθ>θ 1 , with Doppler
and adiabatic terms becoming of comparable importance atθ 1 , and adiabatic
effects dominating at smaller scales. There are various effects that cause the
simple estimate of adiabatic effects to be too large, but they clearly cannot be
neglected forθ<θ 1. A better name, which is starting to gain currency, is the
acoustic peak. In any case, it is clear that the peak is the key diagnostic feature
of the CMB anisotropy spectrum: its height above the SW ‘plateau’ is sensitive
toBand its angular location depends onand. It is therefore no surprise
that many experiments are currently attempting accurate measurements of this
feature. Furthermore, it is apparent that sufficiently accurate experiments will be
able to detect higher ‘harmonics’ of the peak, in the form of smaller oscillations
of amplitude perhaps 20% in power, around"500–1000. These features arise
because the matter–radiation fluid undergoes small-scale oscillations, the phase
of which at last scattering depends on wavelength, since the density oscillation
varies roughly asδ∝exp(icSkτ). Accurate measurement of these oscillations
would pin down the sound speed at last scattering, and help give an independent
measurement of the baryon density.
Since large-scale CMB fluctuations are expected to be dominated by
gravitational potential fluctuations, it was possible to make relatively clear
predictions of the likely level of CMB anisotropies, even in advance of the
first detections. What was required was a measurement of the typical depth of
large-scale potential wells in the universe, and many lines of argument pointed
inevitably to numbers of order 10−^5. This was already clear from the existence of
massive clusters of galaxies with velocity dispersions of up to 1000 km s−^1 :
v^2 ∼GM
r⇒
c^2∼
v^2
c^2,
so the potential well of a cluster is of order 10−^5 deep. More exactly, the
abundance of rich clusters is determined by the amplitudeσ 8 , which measures
[^2 (k)]^1 /^2 at an effective wavenumber of very nearly 0. 17 hMpc−^1. If we assume
that this is a large enough scale so that what we are measuring is the amplitude
of any scale-invariant spectrum, then the earlier expression for the temperature
power spectrum gives
√
TSW^2 10 −^5.^7 σ 8 [g()]−^1.